Fibre-Like Cylinders, Their Packings and Coverings in SL₂R Space

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in SL₂R S L 2 R ~ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups pq2₁ pq 2 1 we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in SL₂R S L 2 R ~ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in H³ H 3 and H²\!× \!R H 2 × R spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of SL₂R S L 2 R ~ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).